A unified approach is presented for establishing a broad class of Cramér–Rao inequalities for the location parameter, including, as special cases, the original inequality of Cramér and Rao, as well as an$L^{p}$version recently established by the authors. The new approach allows for generalized moments and Fisher information measures to be defined by convex functions that are not necessarily homogeneous. In particular, it is shown that associated with any log-concave random variable whose density satisfies certain boundary conditions is a Cramér–Rao inequality for which the given log-concave random variable is the extremal. Applications to specific instances are also provided.

An affine invariant pth moment measure is defined for a random vector and used to prove sharp moment-entropy inequalities that are more general and stronger than standard moment-entropy inequalities.

We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed.

The pth moment matrix is defined for a real random vector, generalizing the classical covariance matrix. Sharp inequalities relating the pth moment and Renyi entropy are established, generalizing the classical inequality relating the second moment and the Shannon entropy. The extremal distributions for these inequalities are completely characterized

The moment-entropy inequality shows that a continuous random variable with given second moment and maximal Shannon entropy must be Gaussian. Stam's inequality shows that a continuous random variable with given Fisher information and minimal Shannon entropy must also be Gaussian. The Crame/spl acute/r-Rao inequality is a direct consequence of these two inequalities. In this paper, the inequalities above are extended to Renyi entropy, p/sup th/ moment, and generalized Fisher information. Generalized Gaussian random densities are introduced and shown to be the extremal densities for the new inequalities. An extension of the Crame/spl acute/r-Rao inequality is derived as a consequence of these moment and Fisher information inequalities.

We show that for a special class of probability distributions that we call contoured distributions, information-theoretic invariants and inequalities are equivalent to geometric invariants and inequalities of bodies in Euclidean space associated with the distributions. Using this, we obtain characterizations of contoured distributions with extremal Shannon and Renyi entropy. We also obtain a new reverse information-theoretic inequality for contoured distributions.